The numerical treatment of nonlinear problems in science and engineering often involves the solution of finite dimensional algebraic systems, or infinite systems in the case of ordinary or partial differential equations. Depending on the specific characteristics and assumptions of the problems, appropriate Newton or Gauss-Newton methods have to be applied, so a large variety of Newton-type methods have been invented over the years.

This monograph covers a multitude of Newton methods and presents the algorithms and their convergence analysis from the perspective of affine invariance, which has been the subject of research by the author since 1970. Four main affine invariance classes are used: affine covariance for error norm controlled algorithms, affine contravariance for residual norm controlled algorithms, affine conjugacy for energy norm controlled algorithms, and affine similarity, which may lead to timestep controlled algorithms. These affine invariance properties are the basis for constructing adaptive Newton methods. Also, the central perspective of affine invariance often leads to different and shorter theorems and proofs.

In the outline of contents, the author thoroughly introduces his central theme of affine invariance and gives a careful overview of the contents of the book. The introduction in the first chapter provides background and basic terms used throughout the book, such as ordinary and simplified Newton methods, Newton-like methods, inexact Newton methods, quasi-Newton methods, Gauss-Newton methods, quasilinearization, and inexact Newton multilevel methods.

The rest of the book is divided into two parts, the first containing chapters 2 to 5, dealing with finite dimensional Newton methods for algebraic equations, and a shorter second part comprising chapters 6 to 8, dealing with infinite dimensional or function space-oriented Newton methods. Chapter 2 is devoted to local Newton methods, for which a sufficiently good initial guess of the solution is provided, and chapter 3 considers global Newton methods for which a sufficiently good initial guess is not assumed. Chapter 4 treats local and global Gauss-Newton methods for nonlinear least squares problems in finite dimensions. Parameter-dependent systems of nonlinear equations are featured in chapter 5. The topics selected for Part 2 include stiff initial value problems for ordinary differential equations and boundary value problems for ordinary and partial differential equations.

The book is intended for graduate students of mathematics and computational science and also for researchers in the area of numerical analysis and scientific computing. Due to the cornucopia of Newton methods provided, the book is a good source for both audiences. As a textbook, it seems to be more suitable for classes than for self-study, since more advanced knowledge of mathematics is required. As a research monograph, the book not only assembles the current state of the art, but also points to future research prospects. The formal mathematical style includes little pseudocode and very few examples. The references are extensive and represent a good selection from the enormous body of literature on Newton methods. Also, there is a good index and a short selection of suitable software packages. In summary, for the more advanced reader, the book is an excellent source about Newton methods, their mathematical foundation, and usage.